We consider a centrosymmetric nonmagnetic crystal with \(N\)  twofold degenerate bands crossing the Fermi level. The Bloch states are degenerate at each wave vector
\( {k} \)  due to the conjugation symmetry \({\cal C}=KI\)  (Ref. Kittel-book): the states \(| {k} ,n,1\rangle \)  and \(| {k} ,n,2\rangle \equiv {\cal C}| {k} ,n,1\rangle \)  are orthogonal and have the same energy.
Here \(K\)  is the time reversal (TR) operation, \(I\)  is the space inversion, \(n=1,...,N\)  labels the bands, and the index \(s=1,2\) , which distinguishes two degenerate states within the same band, is called the
conjugation index. The conjugation operation is antiunitary and, since the Bloch states are spin-\(1/2\)  spinors, we have \({\cal C}| {k} ,n,2\rangle =-| {k} ,n,1\rangle \) , i.e., \({\cal C}^2=-1\) .
The conjugation degeneracy, which is also called the Kramers degeneracy, can be lifted by an external magnetic field. Note that the assumption of inversion symmetry is important: in a non-centrosymmetric crystal
the band degeneracy is lifted by the SO coupling even at zero magnetic field, with wide-reaching implications for superconductivity.[1]}
